Local existence, global existence, and scattering for the nonlinear Schr\"odinger equation

TL;DR

This paper establishes conditions for local and global solutions of the nonlinear Schrödinger equation, including scattering behavior for large initial data, expanding understanding of solution existence and long-term dynamics.

Contribution

It constructs initial value spaces ensuring local solutions for all positive nd complex nd identifies large initial data classes leading to global scattering solutions for xceeding 2/N.

Findings

Existence of local solutions for all nd .

Existence of global scattering solutions for xceeding 2/N.

Construction of initial data spaces for solution analysis.

Abstract

In this paper, we construct for every $\alpha >0$ and $\lambda \in {\mathbb C}$ a space of initial values for which there exists a local solution of the nonlinear Schr\"odinger equation \begin{equation*} \begin{cases} iu_t + \Delta u + \lambda |u|^\alpha u= 0 \\ u(0,x) = u_0 \end{cases} \end{equation*} on ${\mathbb R}^N $. Moreover, we construct for every $\alpha >\frac {2} {N}$ a class of (arbitrarily large) initial values for which there exists a global solution that scatters as $t\to \infty $.